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If x = 3 tant and y = 3 sec t, then the ...

If x = 3 tant and y = 3 sec t, then the value of `(d^2y)/(dx^2)" at" t=pi/4` is

A

`1/6`

B

`1/(6sqrt2)`

C

` 1/(3sqrt2)`

D

` 3/(2sqrt2)`

Text Solution

Verified by Experts

The correct Answer is:
B
We have, x = 3 tan t and y = 3 sec t
`"Clearly",(dy)/(dx) = ((dy)/(dt))/((dx)/(dt)) =(d/(dt)(3 sec t))/(d/(dt)(3 tan t))`
` = (3 sec t tan t )/(3 sec^(2) t) = (tan t)/(sec t) = sin t`
`and (d^(2)y)/(dx^(2))=d/(dx) ((dy)/(dx)) = d/(dt) ((dy)/(dx))*(dt)/(dx) `
`= (d/(dt)((dy)/(dx)))/((dx)/(dt))=(d/(dt)(sin t))/(d/(dt)(3 tan t)) = (cos t )/( 3 sec^(2) t)=(cos^(3)t)/(3) `
Now, `(d^(2)y)/(dx^(2)) ("at "t=pi/4)=(cos^(3)pi/4)/3 = 1/(3(2sqrt2))=1/(6sqrt2)`
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